MTH 312 - 1st Midterm

Definitons, Theorems, & Lemma's For 6.1-6.7

Definition 6.2.1. (Pointwise Convergence)

For each n ∈ N, let fn be a function defined on a set A ⊆ R.

The sequence (fn) of functions converges pointwise on A to a function f if, for

all x ∈ A, the sequence of real numbers fn(x) converges to f(x).

Definition 6.2.3 (Uniform Convergence)

Let (f_n) be a sequence of functions defined on a set A ⊆ R. Then, (f_n) converges uniformly on A to a limit function f defined on A if, for every > 0, there exists an N ∈ N such that |f_n(x) − f(x)| < whenever n ≥ N and x ∈ A.

Definition 6.2.1B.

Let (fn) be a sequence of functions defined on a set A ⊆ R. Then, (fn) converges pointwise on A to a limit f defined on A if, for every > 0 and x ∈ A, there exists an N ∈ N (perhaps dependent on x) such that |f_n(x) − f(x)| < whenever n ≥ N.

Theorem 6.2.5 (Cauchy Criterion for Uniform Convergence)

A sequence of functions (fn) defined on a set A ⊆ R converges uniformly on A if and only if for every > 0 there exists an N ∈ N such that |fn(x) − fm(x)| < whenever m, n ≥ N and x ∈ A.

Theorem 6.2.6 (Continuous Limit Theorem)

Let (fn) be a sequence of functions defined on A ⊆ R that converges uniformly on A to a function f. If each fn is continuous at c ∈ A, then f is continuous at c.

Theorem 6.3.1 (Differentiable Limit Theorem)

Let fn → f pointwise on the closed interval [a, b], and assume that each fn is differentiable. If (f’ n) converges uniformly on [a, b] to a function g, then the function f is differentiable and f’ = g.

Theorem 6.3.2.

Let (fn) be a sequence of differentiable functions defined on the closed interval [a, b], and assume (f’ n) converges uniformly on [a, b]. If there exists a point x_0 ∈ [a, b] where fn(x0) is convergent, then (fn) converges uniformly on [a, b].

Theorem 6.3.3.

Let (f_n) be a sequence of differentiable functions defined on the closed interval [a, b], and assume (f’ _n) converges uniformly to a function g on [a, b]. If there exists a point x₀ ∈ [a, b] for which f_n(x₀) is convergent, then (f_n) converges uniformly. Moreover, the limit function f = lim f_n is differentiable and satisfies f’= g.

Definition 6.4.1.

For each n ∈ N, let fn and f be functions defined on a set A ⊆ R. The infinite series

∑^∞ _n=1 f_n(x) = f₁(x) + f₂(x) + f₃(x) + ···

converges pointwise on A to f(x) if the sequence s_k(x) of partial sums defined by

s_k(x) = f₁(x) + f₂(x) + ··· + f_k(x)

converges pointwise to f(x). The series converges uniformly on A to f if the sequence s_k(x) converges uniformly on A to f(x).
In either case, we write f = ∑^∞ _n=1 f_n or f(x) = ∑^∞ _n=1 f_n(x), always being explicit about the type of convergence involved.

Theorem 6.4.2 (Term-by-term Continuity Theorem)

Let f_n be continuous functions defined on a set A ⊆ R, and assume ∑^∞ _n=1 f_n converges uniformly on A to a function f. Then, f is continuous on A.

Theorem 6.4.3 (Term-by-term Differentiability Theorem)

Let f_n be differentiable functions defined on an interval A, and assume ∑^∞ _n=1 f’_n(x) converges uniformly to a limit g(x) on A. If there exists a point x₀ ∈ [a, b] where ∑^∞ _n=1 f_n(x₀) converges, then the series ∑^∞ _n=1 f_n(x) converges uniformly to a differentiable function f(x) satisfying f’(x) = g(x) on A. In other words,

f(x) = ∑^∞ _n=1 f_n(x) and f (x) = ∑^∞ _n=1 f’_n(x).

Theorem 6.4.4 (Cauchy Criterion for Uniform Convergence of Series)

A series ∑^∞ _n=1 f_n converges uniformly on A ⊆ R if and only if for every ε > 0 there exists an N ∈ N such that

|f_m+1(x) + f_m+2(x) + f_m+3(x) + ··· + f_n(x)| < ε

whenever n > m ≥ N and x ∈ A.

Corollary 6.4.5 (Weierstrass M-Test)

For each n ∈ N, let f_n be a function defined on a set A ⊆ R, and let M_n > 0 be a real number satisfying

|f_n(x)| ≤ M_n

for all x ∈ A. If ∑^∞ _n=1 M_n converges, then ∑^∞ _n=1 f_n converges uniformly on A.

Theorem 6.5.1.

If a power series ∑^∞ _n=0 a_nxⁿ converges at some point x₀ ∈ R, then it converges absolutely for any x satisfying |x| < |x⁰|.

Theorem 6.5.2.

If a power series ∑^∞ _n=0 a_nxⁿ converges absolutely at a point x₀, then it converges uniformly on the closed interval [−c, c], where c = |x₀|.

Lemma 6.5.3 (Abel’s Lemma)

Let b_n satisfy b₁ ≥ b₂ ≥ b₃ ≥ ··· ≥ 0, and let ∑^∞ _n=0 a_n be a series for which the partial sums are bounded. In other words, assume there exists A > 0 such that

|a₁ + a₂ + ··· + a_n| ≤ A

for all n ∈ N. Then, for all n ∈ N,

|a₁b₁ + a₂b₂ + a₃b₃ + ··· + a_nb_n| ≤ Ab₁.

Theorem 6.5.4 (Abel’s Theorem)

Let g(x) = ∑^∞ _n=0 a_nxⁿ be a power series that converges at the point x = R > 0. Then the series converges uniformly on the interval [0, R]. A similar result holds if the series converges at x = −R.

Theorem 6.5.5.

If a power series converges pointwise on the set A ⊆ R, then it converges uniformly on any compact set K ⊆ A.

Theorem 6.5.6.

If ∑^∞ _n=0 a_nxⁿ converges for all x ∈ (−R, R), then the differentiated series ∑^∞ _n=1 na_nxⁿ⁻¹ converges at each x ∈ (−R, R) as well. Consequently, the convergence is uniform on compact sets contained in (−R, R)

Theorem 6.5.7.

Assume f(x) = ∑^∞ _n=0 anxn converges on an interval A ⊆ R. The function f is continuous on A and differentiable on any open interval (−R, R) ⊆ A. The derivative is given by

f’(x) = ∑^∞ _n=1 na_nx_n−1.

Moreover, f is infinitely differentiable on (−R, R), and the successive derivatives can be obtained via term-by-term differentiation of the appropriate series.

Theorem 6.6.2 (Taylor’s Formula)

Let

(3) f(x) = a0 + a1x + a2x2 + a3x3 + a4x4 + a5x5 + ···

be defined on some nontrivial interval centered at zero. Then,

a_n = f⁽ⁿ⁾(0) \ n! .

Theorem 6.6.3 (Lagrange’s Remainder Theorem)

Let f be differentiable N + 1 times on (−R, R), define an = f⁽ⁿ⁾ (0)/n! for n = 0, 1,...,N, and let

S_N(x) = a₀ + a₁x + a₂x² + ··· + a_N x^N .

Given x = 0 in (−R, R), there exists a point c satisfying |c| < |x| where the error function E_N (x) = f(x) − S_N (x) satisfies

E_N (x) = f^(N+1)(c) \ (N + 1)! x^N+1

Theorem 6.7.1 (Weierstrass Approximation Theorem)

Let f : [a, b] → R be continuous. Given ε > 0, there exists a polynomial p(x) satisfying

|f(x) − p(x)| < ε

for all x ∈ [a, b].

Definition 6.7.2.

A continuous function φ : [a, b] → R is polygonal if there is a partition

a = x₀ < x₁ < x₂ < ··· < x_n = b

of [a, b] such that φ is linear on each subinterval [x_i−1, x_i], where i = 1,...n.

Theorem 6.7.3.

Let f : [a, b] → R be continuous. Given ε > 0, there exists a polygonal function φ satisfying

|f(x) − φ(x)| < ε

for all x ∈ [a, b].